Problem: Find $AX$ in the diagram.

[asy]

import markers;

real t=.56;

pair A=(0,0);

pair B=(3,2);

pair C=(.5,1.5);

pair X=t*A+(1-t)*B;

draw(C--A--B--C--X);

label("$A$",A,SW);

label("$B$",B,E);

label("$C$",C,N);

label("$X$",X,SE);

markangle(n=1,radius=15,A,C,X,marker(markinterval(stickframe(n=1),true)));

markangle(n=1,radius=15,X,C,B,marker(markinterval(stickframe(n=1),true)));

label("$24$",.5*(B+X),SE);

label("$28$",.5*(B+C),N);

label("$21$",.5*(A+C),NW);

[/asy]
Answer: The Angle Bisector Theorem tells us that  \[\frac{AC}{AX}=\frac{BC}{BX}\]so \[AX=\frac{AC\cdot BX}{BC}=\frac{21\cdot24}{28}=\frac{7\cdot3\cdot6\cdot 4}{7\cdot4}=\boxed{18}.\]